| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | Particle in hemispherical bowl |
| Difficulty | Standard +0.3 This is a standard circular motion problem requiring resolution of forces (weight and normal reaction) and application of F=mrω². The setup is familiar from M2 syllabi, and while it requires careful geometry and simultaneous equations in part (ii), the techniques are routine for this topic with no novel insight needed. |
| Spec | 3.03d Newton's second law: 2D vectors3.03e Resolve forces: two dimensions6.05c Horizontal circles: conical pendulum, banked tracks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(4 = 10r\) | B1 | \(v = \omega r\), \(r < 0.5\) |
| \(\theta = 53.1°\) | B1 | From \(\sin\theta = 0.4/0.5\) |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(0.4g = 6\cos\theta\) | M1 | |
| \(\theta = 48.2°\) | A1 | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(6\sin\theta = 0.4\omega^2 \times 0.5\sin\theta\) | M1 | \(\text{Accn} = \omega^2 \times 0.5\sin\theta\) |
| (intermediate step) | A1\(\sqrt{}\) | Using cv(48.2°), allow any acute \(\theta\) |
| \(\omega = 5.48 \text{ rad s}^{-1}\) | A1 | |
| Total: 3 |
## Question 4(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $4 = 10r$ | B1 | $v = \omega r$, $r < 0.5$ |
| $\theta = 53.1°$ | B1 | From $\sin\theta = 0.4/0.5$ |
| **Total: 2** | | |
---
## Question 4(ii)(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.4g = 6\cos\theta$ | M1 | |
| $\theta = 48.2°$ | A1 | |
| **Total: 2** | | |
---
## Question 4(ii)(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $6\sin\theta = 0.4\omega^2 \times 0.5\sin\theta$ | M1 | $\text{Accn} = \omega^2 \times 0.5\sin\theta$ |
| (intermediate step) | A1$\sqrt{}$ | Using cv(48.2°), allow any acute $\theta$ |
| $\omega = 5.48 \text{ rad s}^{-1}$ | A1 | |
| **Total: 3** | | |
---4\\
\includegraphics[max width=\textwidth, alt={}, center]{add3948c-3b45-4e67-ac84-e2ca935afd64-05_392_621_255_762}
A particle $P$ of mass 0.4 kg moves with constant speed in a horizontal circle on the smooth inner surface of a fixed hollow hemisphere with centre $O$ and radius 0.5 m (see diagram).\\
(i) Given that the speed of the particle is $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and its angular speed is $10 \mathrm { rad } \mathrm { s } ^ { - 1 }$, calculate the angle between $O P$ and the vertical.\\
(ii) Given instead that the magnitude of the force exerted on $P$ by the hemisphere is 6 N , calculate
\begin{enumerate}[label=(\alph*)]
\item the angle between $O P$ and the vertical,
\item the angular speed of $P$.
\end{enumerate}
\hfill \mbox{\textit{CAIE M2 Q4 [7]}}